Degree Sequence

Definition

A degree sequence is a list of the degrees of the vertices in a graph or hypergraph, typically arranged in non-increasing order. This sequence provides crucial information about the structure and properties of the graph, influencing various combinatorial and extremal considerations, including connectivity and the existence of certain subgraphs.

5 Must Know Facts For Your Next Test

1. The degree sequence can reveal important properties about a graph, such as whether it can be connected or contains cycles.
2. In extremal combinatorics, the degree sequence is often analyzed to find upper and lower bounds on the number of edges a graph can have given certain constraints.
3. A degree sequence must satisfy the Handshaking Lemma, which states that the sum of the degrees of all vertices must be even.
4. For hypergraphs, the concept of a degree sequence extends to include the size of hyperedges, affecting the structure and properties of the hypergraph.
5. Not all sequences of integers can be realized as a degree sequence; certain conditions must be met, such as being graphical or k-graphical depending on the type of graph or hypergraph.

Review Questions

• How does the degree sequence relate to the structural properties of graphs?
  • The degree sequence directly impacts several structural properties of graphs, including connectivity and the presence of specific subgraphs. For example, if a degree sequence indicates that there are many high-degree vertices, it suggests that those vertices are well-connected and can lead to a more interconnected graph. Conversely, a low-degree sequence may point to potential isolated vertices or disconnected components within the graph.
• Discuss the significance of the Havel-Hakimi Algorithm in determining whether a degree sequence can form a simple graph.
  • The Havel-Hakimi Algorithm plays a crucial role in validating whether a given degree sequence can correspond to a simple graph. It systematically reduces the sequence by removing the largest degree and connecting that vertex to other vertices based on their remaining degrees. If at any step, a negative degree arises or if not all degrees can be satisfied by connections, it indicates that the original degree sequence cannot form a simple graph. This process is vital for understanding how degree sequences influence graph construction.
• Evaluate how degree sequences affect extremal problems in both graphs and hypergraphs, and provide examples of these applications.
  • Degree sequences are fundamental in extremal problems as they help define constraints on how many edges or hyperedges can exist given specific conditions. For instance, in a graph with a fixed degree sequence, one might want to maximize or minimize certain substructures while adhering to those degrees. In hypergraphs, analyzing degree sequences can lead to insights into covering problems or finding certain configurations that satisfy given parameters. These applications highlight the importance of understanding how variations in degree sequences can dramatically influence combinatorial properties.

Related terms

Graph: A collection of vertices connected by edges, representing relationships between pairs of objects.

Hypergraph:

A generalization of a graph where an edge can connect any number of vertices, allowing for more complex relationships.

Havel-Hakimi Algorithm: A constructive algorithm used to determine if a given degree sequence can represent a simple graph.